String Girdling Earth Puzzle

Calculate the gap created by adding length to a string wrapped around a sphere, or find the required length for a specific gap.

This tool helps you solve the classic mathematical brain teaser about wrapping a string around the Earth. Enter the length you want to add to see how high the string will lift off the surface, or enter a desired height to find out how much extra string you'll need. The result might surprise you!

Practical Examples

Load an example to see how the calculator works.

The Classic 1 Meter Problem

Find Gap Height

Calculate the gap if you add 1 meter to a string tightly wrapped around the Earth's equator.

Added Length: 1 m

Adding 6 Feet of String

Find Gap Height

If you add a 6-foot section of string (about a person's height), how much does the string rise?

Added Length: 6 m

Achieving a 1 Foot Gap

Find Added Length

How much extra string do you need to add to make it hover 1 foot off the ground?

Gap Height: 1 m

A Car-Sized Gap

Find Added Length

How much extra string is needed for a car (approx. 1.5m high) to pass under it?

Gap Height: 1.5 m

Other Titles
Understanding the String Girdling Earth Problem: A Comprehensive Guide
An in-depth look at the famous mathematical riddle, its surprising solution, and its implications.

What is the String Girdling Earth Problem?

  • The Classic Riddle
  • The Counter-Intuitive Solution
  • Why It Works: The Math Behind the Magic
The String Girdling Earth problem is a classic mathematical brain teaser that demonstrates a fascinating and counter-intuitive geometric result. It's often presented as a thought experiment that challenges our assumptions about scale and proportion.
The Classic Riddle
Imagine you have a string that is just long enough to wrap perfectly around the Earth's equator, touching the ground at every point. Now, let's say you add just 1 meter of length to this string. If you were to lift this now slightly longer string off the ground so that it forms a perfect circle, maintaining a uniform height above the equator, how big would the gap be between the string and the ground? Could a piece of paper slide under it? A mouse? Could you crawl under it?
The Counter-Intuitive Solution
Most people's intuition suggests that with the Earth being so massive, adding a mere 1 meter to a string that's already ~40,000 kilometers long would result in a minuscule, almost unnoticeable gap. The surprising answer is that the gap would be approximately 15.9 centimeters (or about 6.3 inches) high all the way around. That's high enough for a small animal to easily pass under! The most astonishing part is that this result is true for any sphere, whether it's the size of a tennis ball or a star like the Sun. The radius of the object doesn't matter.
Why It Works: The Math Behind the Magic
The solution lies in the simple formula for the circumference of a circle, C = 2πr, where 'C' is the circumference, 'r' is the radius, and 'π' (pi) is the mathematical constant approximately equal to 3.14159. When we add length to the string, we are creating a new, larger circle with a new circumference and a new radius. The gap is the difference between the new radius and the original radius. The math shows that this difference, the gap height (h), depends only on the added length (L), through the formula h = L / 2π. The original radius cancels out of the equation entirely.

Puzzle Examples

  • A string around the Earth's equator is made 1 meter longer. How high does it float? Answer: ~16cm
  • The result is independent of the initial sphere's radius
  • Illustrates that C = 2πr, so ΔC = 2πΔr
  • A classic mathematical brain teaser

Step-by-Step Guide to Using the String Girdling Earth Calculator

  • Choosing Your Calculation
  • Entering Your Values
  • Interpreting the Results
Our calculator simplifies this problem, allowing you to explore the relationship between added length and gap height without manual calculations.
Choosing Your Calculation

First, use the 'Calculation Type' dropdown to select what you want to find. You have two options:

  1. Find Gap Height: Use this if you know how much extra string you're adding (the 'Added Length') and want to find the resulting height of the gap.
  2. Find Added Length: Use this if you have a specific gap height in mind and want to know how much extra string you'd need to achieve it.
Entering Your Values
Based on your selection, an input field will appear. For 'Find Gap Height', you'll enter the 'Added Length (L)'. For 'Find Added Length', you'll enter the 'Gap Height (h)'. Ensure you enter a positive number. The units are consistent; if you enter the length in meters, the resulting gap will be in meters.
Interpreting the Results
After clicking 'Calculate', the result will be displayed instantly. If you calculated the gap height, it will show the uniform distance between the sphere's surface and the string. If you calculated the added length, it will show the total extra string needed to create your desired gap.

Mathematical Derivation and Formulas

  • Deriving the Gap Height Formula
  • Deriving the Added Length Formula
  • The Independence of the Radius
The logic behind the calculator is based on fundamental geometric principles. Let's walk through the derivation.
Deriving the Gap Height (h) Formula
  1. Let R be the original radius of the sphere (e.g., the Earth).
  2. The original circumference is C₁ = 2πR.
  3. Let L be the length of string added. The new circumference is C₂ = C₁ + L = 2πR + L.
  4. The new circumference corresponds to a new, larger radius, let's call it Rnew. So, C₂ = 2πRnew.
  5. Setting the two expressions for C₂ equal: 2πR_new = 2πR + L.
  6. To find the new radius, divide the entire equation by 2π: R_new = R + L/2π.
  7. The gap height (h) is the difference between the new radius and the old radius: h = R_new - R.
  8. Substituting the expression for R_new: h = (R + L/2π) - R.
  9. This simplifies to: h = L / 2π.
Deriving the Added Length (L) Formula
We can just as easily work backward. If we know the desired gap height (h), we can rearrange the formula to solve for the required added length (L).
  1. Start with the gap height formula: h = L / 2π.
  2. To isolate L, multiply both sides by 2π.
  3. This gives us the formula: L = h * 2π.
The Independence of the Radius
Notice how in the final formula, h = L / 2π, the variable 'R' for the original radius has completely disappeared. This is the mathematical proof that the size of the initial sphere is irrelevant to the final gap height. The relationship between the change in circumference and the change in radius is constant, governed only by the factor of 2π.

Real-World Applications and Analogies

  • Engineering and Manufacturing Tolerances
  • Orbital Mechanics
  • A Lesson in Proportionality
While a string around the Earth is a thought experiment, the principle it illustrates has tangible applications in science and engineering.
Engineering and Manufacturing Tolerances
In manufacturing, especially with circular or cylindrical parts like pipes, bearings, or rings, this principle is crucial. A small change in the required circumference (or perimeter) of a part leads to a predictable change in its radius or diameter. Engineers use this relationship (Δr = ΔC / 2π) to set manufacturing tolerances. If a pipe's circumference must be within a certain range, they can calculate the corresponding acceptable range for its radius or diameter.
Orbital Mechanics
The problem is analogous to changes in satellite orbits. While orbits are governed by gravity and are more complex (often elliptical), a similar principle applies. To raise a satellite to a higher circular orbit (a larger radius), you must increase its speed, which in turn increases the path length (circumference) it travels. The relationship between the change in orbital altitude (the 'gap') and the change in orbital path length is conceptually similar to the string problem.
A Lesson in Proportionality
Fundamentally, the riddle is a powerful lesson in linear relationships and proportionality. It teaches us that for all circles, the circumference and radius are directly proportional. Any change in one produces a proportional change in the other, and the constant of proportionality is always 2π. This is a core concept in geometry, trigonometry, and physics.

Common Questions and Further Exploration

  • What if the object isn't a perfect sphere?
  • What about the weight or sag of the string?
  • Exploring different shapes
The classic problem makes a few simplifying assumptions. Let's explore what happens when we question them.
What if the object isn't a perfect sphere?
The Earth is not a perfect sphere; it's an 'oblate spheroid,' slightly wider at the equator than from pole to pole. For an irregular, non-circular shape, the concept of a uniform gap becomes complex. You would still be adding length to the perimeter, but the 'gap' would vary depending on the local curvature of the surface. However, the average gap would still be conceptually similar.
What about the weight or sag of the string?
The riddle assumes a magical, weightless string that can maintain a perfect circular shape without support. In reality, a physical string would sag due to gravity. To maintain a uniform height, it would need to be supported at countless points or be held under immense tension, which would stretch it further. The problem is a purely geometric one and ignores these real-world physics.
Exploring different shapes
A fascinating extension of this problem is to consider a string wrapped around a cube. If you add length to a string wrapped around the 'equator' of a cube and pull it out to form a shape that is 'uniformly' distant, what happens? The new shape would be a circle, and the gap would be largest at the midpoint of the cube's faces and smallest at the corners, where the string would still touch. This variation highlights how the principle is specifically about circles and constant curvature.